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Re(•) Real part of the complex-valued variable • Observability (§2.1.1) Redundancy (§2.1.2) 1 Introduction Accuracy (§2.1.3) Sensing and measurement systems are ubiquitous in electric power grids. Dynamic state estimation (§2.2) From power generation stations to end-customer sites, measurement sys- tems are continuously acquiring raw data that are mission-critical to the Observability (§2.2.1) long-term planning and real-time monitoring and control of power grids. In long-term planning, recorded measurements are used for model val- Stability assessment and control (§2.3) idation and calibration [1] as well as for model reduction [2]. Further, recorded measurements are used for the postmortem analysis of major Voltage (§2.3.1) events such as blackouts [3]. As for monitoring and control, sampled Rotor angle (§2.3.2) measurements are continuously feeding energy management systems running at control centers [4] across the country. Sampled measure- Topology change detection (§2.4) ments enable near-real-time situational awareness, and corrective control actions are taken based on the analysis of the available information. Fault detection (§2.5) Examples of the application for monitoring and control include static state estimation [5], security assessment [6, 7], event detection [8], and Power quality monitoring (§2.6) voltage control [9]. Distribution (§3) Despite their extension, experts recognize the need for expanding the measurement systems further in the near future [10]. One reason lies in Static state estimation (§3.1) the advent of new technologies—not only in sensing and measurement technology but also in communications and networking, optimization Observability (§3.1.1) and control, energy conversion and storage, and power electronics—that are transforming the electric energy sector. The ongoing developments Stability assessment and control (§3.2) are being referred to as grid modernization, and they are mainly driven by governmental initiatives aimed at the safe and reliable electrification Voltage (§3.2.1) of the economy. The Grid Modernization Laboratory Consortium∗, sup- Topology change detection (§3.3) ported by the U.S. Department of Energy, and the Global Power System Transformation Consortium∗∗, are examples of such initiatives. Fault detection (§3.4) Grid modernization initiatives are pushing for rapid growth in the adoption of solar and wind power plants as primary sources of electric Power quality monitoring (§3.5) energy. Despite their advantages, however, these sources are naturally variable and stochastic [11]. As a consequence, empirical data show Fig. 1: Categorization by application of the measurement placement that system states are experiencing excursions more often and more problem in electric power grids. abruptly than in the past. Advanced controls are required to accommo- date this variability and intermittency while maintaining grid reliability and enhancing grid resilience; in turn, additional measurements at higher spatial and temporal resolutions are required for advanced controls. This is arguably a worldwide trend. Inform the industry, policymakers, and researchers on the research- In this scenario of the continuous expansion of measurement systems, •and-development needs for measurement placement. the following question is posed: Can the synergies across disparate measurement systems be exploited Attaining these objectives is obviously challenging. First, the litera- for an optimal placement solution that serves multiple applications? ture on this topic is vast; chronologically organizing key contributions on In pursuit of the answers, this paper reviews the prominent applica- this topic is a laborious task. Second, the synergy among existing strate- tions for which measurement placement strategies have been formulated. gies is not obvious. To begin with, measurement placement is a classic The paper also delves into the relatively newer applications for which planning problem, and it is challenging because the usage of measure- measurement placement strategies have been formulated, particularly in ments covers almost all the different types of power system research the distribution grid because of the continuous increase in the number of and analysis. Nonetheless, a multi-objective framework that offers a distributed energy resources (DERs). The objective is to fulfill a gap in parsimonious solution to the contemporary problem of measurement the open literature for a systematic framework guiding the placement of placement for the future grid modernization scenarios and applications is disparate measurement systems synergistically. required. The solution must recognize that future grids will include more This study is motivated by the fact that the current practice in the uncertainty. Third, the identification of research-and-development needs industry hinges heavily on engineering judgment and the analysis of that are worth the time and investment cost requires cross-disciplinary worst-case scenarios despite the extensive literature on the formulation expertise. of measurement placement strategies. For example, some of the most The reviewed measurement placement methods are separated into common industry practices include placing measurements to monitor two sections—one for transmission grids and the other for distribu- critical devices/sites (e.g., step-up transformers in large power stations tion grids—to effectively address these challenges. See Fig. 1. The and tie-line buses) and to gather further information on specific loca- methods are categorized within these sections by application (and sub- tions to fix localized problems such as voltage stability [12]. On the other application, in some cases). For example, §2.1.2 presents and discusses hand, academic publications tend to focus on specific applications. For methods aimed at enhancing the measurement redundancy for static state example, there is much literature on the optimal placement of measure- estimation in transmission grids. This categorization is critical in review- ments to attain observability and thus enable static state estimation in ing such extensive literature without losing sight of the main objectives transmission grids [13]. Of course, these approaches are well-justified previously outlined. within their own context. Industry needs a method that is reliable and Note that the present work aims to provide insight into the works that cost-efficient yet simple, whereas academia mostly seeks novelty and made fundamentally new contributions to the literature or exposed an rigor in the problem formulation. Accordingly, the primary role of this important factor to the problem of measurement placement; incremental paper is to build a bridge between industry practices and existing formu- contributions are referred to without further elaboration. Therefore, the lations of the measurement placement problem. Further, this paper aims References section is not exhaustive—full cataloging of the existing lit- to review, present, and discuss measurement placement formulations in erature on the present topic is beyond the scope of this paper. Second, an accessible yet rigorous form. From a broader perspective, however, before proceeding, we stress that many measurement systems acquiring this paper serves multiple objectives including to: nonelectrical quantities are found throughout electric grids. For example, Review, present, and discuss existing formulations for measurement equipment-level measurements of dissolved gas, vibration, temperature, placement• in power grids. The review is thorough and rigorous though pressure, humidity, and strain are used for equipment diagnostics. Mea- accessible to the larger community. surements of solar irradiance and wind speed integrated with solar and Identify synergies among different strategies that can lead to improved wind power plants, respectively, are used in operational planning. Binary solutions.• measurements of switchgear status are used for monitoring, topology processing, and control. In this paper, we focus primarily on measure- ment systems that acquire electrical quantities; within these, we are specifically interested in the measurements that support monitoring and control at a systems level. Table 1 summarizes the measurement sys- ∗https://gmlc.doe.gov/ tems of interest. Note that sample values acquired by merging units are ∗∗https://globalpst.org/ popularly known in the United States as point-on-wave measurements.

pp. 1–18 2 Table 1 Considered measurement systems and their location in the grid WEIGHTED LEAST SQUARES (WLS) ESTIMATOR

A state estimate, x, is obtained by minimizing an objective function of choice, J, as follows: x = argmin J(x). (2) b x The WLS estimator minimizes a quadratic criterion:

Generation site 1–25 kV Transmission 69–1,000 kV Subtransmission 15–69 kV Distribution 7.2–15 kV Customer site 100–240 V b 1 T 1 J(x)= (z Hx) R− (z Hx). (3) Smart meter X 2 − − Power quality monitor X X The solution, x, is obtained by setting to zero the partial derivative Digital fault recorder X X X X of J(x) with respect to x: Digital protective relay X X X X PMU X X X ∂J(x) b T 1 T 1 x = = H R− (z Hx)= H R− r = 0, (4) Micro-PMU X ∂x − − − X X x=x Merging unit [14, 15] where the residual vector r = z Hx. From (4), we have: b b b T − T H R 1z + H R 1Hx = 0, (5) yielding to: − − − 1 b The most frequently adopted strategies to formulate the measurement T 1 − T 1 1 T 1 x = H R− H H R− z = G− H R− z. (6) placement problem include: b Minimize the total cost, including measurement devices and systems, From (6), if the gain matrix, G, is nonsingular, then (1) is •infrastructure, data communications, storage, and processing. numericallyb observable. Recall that R is a diagonal matrix. Ensure system observability under normal and anomalous operating conditions.• Maximize the performance of the applications using the measure- • ments, e.g., state estimation accuracy. 6 PMU The first factor models the investment cost elements, whereas the iii vi other two capture technical aspects. The importance of including cost in the problem formulation is illustrated by an anecdotal example. Sup- 3 pose that the operator of a power transmission grid of 900 buses decides i ii v vii to invest in making the measurement system observable solely by pha- sor measurement units (PMUs). The details of this problem, including 1 2 4 5 the definition of observability, will be given in the next section. For iv viii now, it is sufficient to know that at least 300 PMUs are needed [16]. Based on a report for the U.S. Department of Energy [17], the cost of 7 a PMU—including procurement, installation, and commissioning—can Fig. 2: One-line diagram of the 7-bus system [24]. The bus (branch) range from $40,000–$180,000, depending on the device class of pre- indices are indicated by Arabic (Roman) numerals. cision, the number of measurement channels, and other characteristics; hence, the total investment in this anecdotal example ranges from $12 million–$54 million. Note that it is not uncommon to find transmission grids of this size around the world. The magnitude of the measurement OBSERVABILITY FOR STATIC STATE ESTIMATION IN placement in electric power grids—in terms of both problem dimension TRANSMISSION GRIDS and involved costs—is formidable. Additional technical factors will be uncovered next. The key contributions to power system observability analysis The paper proceeds as follows. Sections II and III present mea- are attributed to Clements, Krumpholz, and Davis [25, 26]. Two surement placement strategies applied to electric power transmission and distribution grids, respectively. Section IV provides an outlook on definitions of observability are well accepted: what has been accomplished and delineates the major future research Numerical observability is defined as the ability of the measure- directions in this field. Section V concludes the paper. ment model (1) to be solved for a state estimate x. If H in (1) is of full rank and well conditioned, or equivalently, if G in (6) is nonsingular, then the system is said to be numerically observable. 2 Measurement placement in transmission grids b In legacy transmission grids, all major substations are equipped with Topological observability is defined as the existence of at least remote terminal units (RTUs); their role is to gather the measurements one spanning measurement tree of full rank in the network. collected at the substation and transmit them to centralized control cen- Numerical observability implies topological observability, but the ters. Typically, RTUs transmit a batch of measurements once every 1–4 converse is not true. In practice, however, cases where a power grid seconds. In this paper, the placement of RTUs and associated sensing and is topologically but not numerically observable are rare. See, e.g., measurement devices is not considered. This is in part because another [27] for more details. device, based on more recent technology and referred to as PMU, has taken precedence in transmission grids. Depending on their class and manufacturer, PMUs can transmit a batch of Global Positioning System- synchronized measurements up to 240 times per second [18]. On a historical note, PMUs became commercially available a few years after estimator [see the box Weighted least squares (WLS) estimator] is the the development of the prototype was completed in 1988 [19, 20]. The maximum likelihood estimator under these conditions. Note that the U.S. Department of Energy released the wide-area measurement system measurement model (1) is linear because PMUs measure both the magni- project [21, 22] shortly after that. The placement of PMUs has been tude and phase angle of voltages and currents [23]. For example, suppose well explored since then, with initial work dating to the early 1990s. A that buses 2,4,6,7 in the network shown in Fig. 2 are equipped with review of PMU placement methodologies developed up to 2011, with a PMUs. In{ this case,} the measurement model is given explicitly in (7), jθ focus on static state estimation only, is given in [13]. In a complementary where vk = vk e k , vk (θk) is the voltage magnitude (phase angle) at fashion, this paper encompasses all the different applications that can be bus k;v ˜k (vk|) denotes| | the| measured (true) algebraic state variable asso- jαk considered for PMU placement, starting with static state estimation. ciated with bus k; ikℓ = ikℓ e ℓ , ikℓ (αkℓ) is the current magnitude (phase angle) from bus k|to |ℓ; i˜ denotes| | the measured current phasor 2.1 Static state estimation in transmission grids kℓ from bus k to ℓ; yk0 is the shunt admittance of bus k; and ykℓ is the series admittance between buses k and ℓ. Hereafter, unless otherwise stated, it Consider an electric power network of nbu buses, equipped with m mea- is assumed that a PMU installed on bus k measures the voltage phasor at surements contained in the measurement vector, z. The measurement bus k as well as the current phasor in all the branches that are incident to model is given by: bus k. For example, if a PMU is placed on Bus 2 of the network in Fig. 2, z = Hx + e, (1) then measurements of v2,i21,i23,i26,i27 are supposedly available from { } where H is the measurement matrix, and x is the true algebraic state this device. In other words, the number of measurement channels in each vector. The elements of the measurement error vector, e, are assumed to PMU is assumed to be unlimited unless otherwise stated. be zero-mean, independent and identically distributed random processes e N 0 R 2.1.1 Observability for static state estimation in transmission following a Gaussian probability density function, ( , )— that grids: is, E (e) = 0, E (eeT)= R = diag(σ 2,...,σ 2), where∼ E denotes the 1 m Minimum PMU placement for observability in ideal conditions: expectation operator, R is the error covariance matrix, and σk is the standard deviation associated with k-th measurement zk. This is a conve- The most basic requirement for state estimation is the observability nient assumption because the widely used weighted least squares (WLS) of the measurement model (1) [see the box Observability for static state

pp. 1–18 3 v˜2 0 10 00 0 0 e1 v˜4 v˜ 0 00 10 0 0 e2  6  0 00 00 1 0 v˜7    e3  ˜ 0 00 00 0 1 e4  i21  ˜ y12 y20 + y12 0 00 0 0 e5  i23    v1      − 0 y20 + y23 y23 00 0 0   e6   i˜26   −  v2    ˜   0 y20 + y26 0 0 0 y26 0  v3  e7   i27   0 y + y 0 00− 0 y  v4  e  (7)  ˜  =  20 27 27   +  8 .  i43   0 0 y34 y40 + y34 0 0− 0  v5  e9   ˜   −  v6    i45   0 0 0 y40 + y45 y45 0 0  v   e10   i˜   0 0 0 y + y − 0 0 y  7   e   47   40 47 47    11  i˜ 0 y26 0 0 0 y60 + y26 − 0 e12  62   −     i˜   0 0 y36 0 0 y60 + y36 0   e13   63   −     i˜   0 y27 0 00 0 y70 + y27   e14   72   −   e   i˜   0 0 0 y47 0 0 y70 + y47   15   74   −      estimation in transmission grids]. This is because H is assumed to be ADMITTANCE MATRIX AND CONNECTIVITY MATRICES perfectly known; thus, if (1) is observable, one can rely on the measure- ments in z to obtain an estimate, x, of the algebraic state vector. On the other hand, installing a PMU on each bus of an electric transmission The admittance matrix—also called Ybus or Y matrix or bus grid is cost-prohibitive; hence, a relevant question to ask is, what is the admittance matrix or nodal admittance matrix—is a matrix of minimum set of PMUs that makesb the measurement model (1) observ- dimension nbu nbu that represents the nodal admittance of electric able? This question is investigated in [16, 28] by extending the notion power networks.× The Ybus is likely the most used matrix in power of a spanning measurement tree to the specific case of PMUs. Following system studies, and it has several interesting properties. For example, [16], a spanning measurement tree is any network subgraph that “con- the Ybus is: tains all the nodes of the network and has an actual measurement or Very sparse for real electric power transmission grids; a calculated pseudo-measurement assigned to each of its branches. A • Symmetric if no phase-shifting transformers are included. pseudo-measurement is assigned to either a non-metered branch where • For the network shown in Fig. 2: the voltage phasor at both ends are known [using Ohm’s law], or to a non-metered branch which is incident to a bus where all but the current y11 y12 0 0 0 0 0 of that branch are known [using Kirchhoff’s current law].” In particular, y12 −y22 y23 0 0 y26 y27 − − − − it is shown in [16] through numerical simulations performed on various  0 y23 y33 y34 0 y36 0  test systems that approximately: Y = 0− 0 y34 −y44 y45 − 0 y47 . One-fourth to one-third of the network buses in general and  0 0− 0 y45 −y55 0− 0   −  • One-half of the network buses in the worst-case scenario  0 y26 y36 0 0 y66 0  •  0 −y27 − 0 y47 0 0 y77  need to be instrumented with PMUs to achieve observability. The  − −  pioneering work in [16, 28] sets the ground for the minimum PMU One can use Y , which is readily available, to build the bus-to-bus placement problem in transmission grids. It relies on a dual search connectivity matrix, A. To do so, simply set all nonzero elements algorithm, comprising a modified bisecting search and a simulated of Y to 1. Also, A can be built by using (11). annealing method, to build the spanning measurement tree of the net- work. When computing time is of concern, several modifications can be The bus-to-branch connectivity matrix, denoted by B in (23), applied to the heuristics in [16] to accelerate the solution process [29]. can be built by using (22). See, e.g., [56] for more details. See also [30] for some variations of the heuristics in [16] and [31] for a method based on Tabu search. The reader interested in a formal treatment of the minimum PMU placement problem is referred to [32, 33], where a graph-theoretic approach is taken to prove that, under an ideal scenario, no more than one-third of the network buses need to be provided with PMUs for 1 if a PMU is installed on bus k, u = (9) observability—we stress that this holds only under ideal scenarios. In k 0 otherwise. fact, we discuss a counterexample later in this section where more than  one-half of the network buses need to be instrumented with PMUs to The solution to the optimization problem in (8) is obtained by inte- achieve observability. It is also shown in [32, 33] that the minimum PMU ger programming (IP). For this reason, (8) is hereafter referred to as the placement problem is NP-complete. The metaheuristics used to find a IP method. During the solution process, the inequality constraints asso- solution to the problem under discussion are summarized in Table 2. ciated with g(u) enforce topological observability. The rationale of the mechanism by which topological observability is enforced is explained next. Consider: nbu Table 2 Metaheuristics used to find the minimum set of PMUs for observability gk(u)= ∑ Akℓ uℓ 1, (10) Metaheuristic Ref. Metaheuristic Ref. ℓ=1 · ≥ Bisecting search [16, 28] Fuzzy logic [34] where: Cellular learning automata [35] Genetic algorithm [36–41] 1 if k = ℓ, or if buses k and ℓ are connected, Chemical reaction optimization [42] Iterated local search [43] Akℓ = (11) Cuckoo search algorithm [44] Particle swarm optimization [45–48] 0 otherwise, Evolutionary algorithm [49–51] Simulated annealing [16, 28, 29, 52, 53]  Exhaustive binary search [54] Tabu search [29, 31, 55] denotes the kℓ-th element of the bus-to-bus connectivity matrix [see the box Admittance matrix and connectivity matrices]. The constraints defined in (10) are better explained through an example. To this end, consider again the network in Fig. 2. From (10)–(11), we have: Note, reference [54] in Table 2 uses an exhaustive binary search to g1 = u1 + u2 1, find the minimum number of PMUs that makes (1) topologically observ- 1100000 ≥ able. Despite being computationally expensive, the exhaustive search 1110011 g2 = u1 + u2 + u3 + u6 + u7 1, ≥ provides a globally optimal solution; thus, the results obtained for the  0111010   g3 = u2 + u3 + u4 + u6 1,  ≥ systems in [54]—IEEE 14-bus, IEEE 24-bus, IEEE 30-bus, and New A = 0011101 , and  g4 = u3 + u4 + u5 + u7 1, England 39-bus—serve as a benchmark solution. Other than an exhaus-    ≥  0001100   g5 = u4 + u5 1, tive search, the metaheuristics in Table 2 do not guarantee a globally  0110010   ≥   g6 = u2 + u3 + u6 1, optimal solution. Unfortunately, an exhaustive search becomes quickly  0101001  ≥ impractical as the number of buses (and thereby the dimension of the    g7 = u2 + u4 + u7 1.    ≥ (12) problem) increases because the problem is NP-complete.  Instead of a heuristic, the formulation in [24] relies on an optimization The constraints in (12) guarantee topological observability by enforc- model, as follows: n ing that at least one uℓ = 1 for each gk, k,ℓ = (1,...,nbu). Or, in plain bu words, they guarantee that the voltage phasor on each bus in the network min ck uk, ∑ · is either directly measured or indirectly calculated. The IP method is well k=1 (8) accepted because of its simplicity and scalability. The reader seeking s.t. g(u) 1n , an in-depth understanding should start from [57] and references therein. ≥ bu The IP method can be modified to consider measurements acquired from Z 1 where ck denotes the cost of installing a PMU on bus k, gk(u) , nbu RTUs and/or pseudo-measurements inferred from zero-injection buses. denotes the vector of all ones of dimension nbu, and: ∈ This is accomplished by modifying the constraints associated with g(u).

pp. 1–18 4 Note that a zero-injection bus is a bus without generators, loads, or any of the last stage, the number of installed PMUs will be larger than the other shunt device connected to it. For example, let Bus 3 in the 7-bus minimum necessary for observability. In other words, this approach leads system shown in Fig. 2 be a zero-injection bus. If the voltage phasor in to a larger number of placed PMUs than [16], [24], and others. Nonethe- any three buses in the set 2,3,4,6 are measured, then the voltage pha- less, the idea of installing PMUs in a phased manner is of practical sor at the fourth bus can{ be calculated} by applying Kirchhoff’s current interest given the amount of investment involved. Along these lines, let law at Bus 3, where the net injected current is known. Following [24], S, card(S)= np, be the minimum set of PMUs required to attain observ- the constraints in (12) are modified as follows: ability. Ideally, card(S)= card(S1 S2 St ), where St denotes the g2 = u1 + u2 + u3 + u6 + u7 + g3 g4 g6 1, · · ≥ set of PMUs installed in the last stage,∪ ∪t. ··· This ∪ condition would ensure g4 = u3 + u4 + u5 + u7 + g2 g3 g6 1, (13) that upon completion of the last stage, the number of PMUs installed is g6 = u2 + u3 + u6 + g2 g3 g· 4 · 1. ≥ · · ≥ not larger than the minimum set of PMUs required to attain observability. By replacing g2, g3, g4, and g6 into (13) and simplifying via Boolean On the other hand, the measurement model will not be observable until logic, one obtains: the last stage is completed. This idea is pursued in [69], which accounts g2 = u1 + u2 + u3 + u6 + u7 1, for an important extension of the IP method. The minimum set of PMUs g4 = u2 + u3 + u4 + u5 + u6 +≥u7 1, (14) required to attain observability is determined beforehand. Also, the num- g6 = u2 + u3 + u6 + u1 u4 + u4 u≥7 1. ber of PMUs to be installed at each stage is defined a priori based on, e.g., · · ≥ the available budget for each stage. Then, for each stage ℓ: Note that the modification suggested in [24] makes the constraints nbu in (14) nonlinear, which is not desirable. A better approach to modify max ∑ sk, the constraints in (8) is developed in [58, 59] and extended in [60]. The k=1 interested reader is referred to those papers for details. See also [61–65]. s.t. Au s, Note that considering zero-injection buses is desirable because it reduces ≥ the number of PMUs required for system observability. The IP method 0 k S, (16) uk = ∀ 6∈ can also be modified to account for the single loss of any PMU [66] 1 k S0 S1 St 1, ∀ ∈ ∪ ∪ ···∪ − and other relevant factors. Extensions of the IP method will be discussed nbu  t in several opportunities throughout the paper. The trade-offs between ∑ uk = ∑ nℓ. some of these extensions and the obtained solutions are well presented k 1 1 in [62, 67]. Specifically, in [62], the set of constraints in the IP method = ℓ= is extended such that: The interpretation of (16) is simple. The set S is known, but not all PMUs can be installed at once. The question is what is the best choice of Zero-injection buses are considered for PMU installation. S S, ℓ = 1,...,t. The strategy in (16) is, for each stage ℓ, to maximize • The system remains observable in case of the loss of any single PMU. ℓ ⊂ • The system remains observable in case of single branch outages. the number of buses in which the voltage phasor is measured or can be • The number of measurement channels in some or all PMUs is limited. calculated, given the number of PMUs to be installed at stage ℓ, nℓ < np. • In (16), sk is equal to 1 if the voltage at bus k is directly measured or can Moreover, [62] suggests a specific optimization solver in which an be calculated, and 0 otherwise. The inequality Au s is a relaxation of optimality gap can be specified as a trade-off between a slower/optimal (10). The equality u = 0 k S ensures that only elements≥ of S are can- and a faster/suboptimal solution. k didates for placement, whereas∀ 6∈ uk = 1 k S0 S1 St 1 fixes the − In practice, the deployment of, e.g., nbu/3 PMUs, requires an enor- candidates chosen in earlier stages; S0 ∀= ∅∈. The∪ last∪···∪ equality constraint mous effort, even for limited-size networks. Considering this practical restricts the number of PMUs allowed to be placed at stage ℓ. The for- aspect, the work in [52, 53] develops the concept of degree of unob- mulation in (16) is amenable to the consideration of zero-injection buses, servability, which is better explained through an example. Consider the and the constraints remain linear, as is the case in [58, 59]; therefore, network in Fig. 3. In Placement 1, the following quantities are measured: this formulation is henceforth referred to as the multistage ILP method. v2,i21,i23,v6,i65,i67 ; thus, the voltage phasor at buses 2,6 is directly In addition to (16), the work in [69] develops a simple yet effective way {measured by PMUs,} and the voltage phasor at buses 1{,3,5},7 can be of including a measure of the degree of redundancy in the optimization calculated by using the following relation: { } model. The number of PMUs that provides access to the voltage phasor at bus k is: i21 y20 + y12 0 y12 0 0 0 v1 (k) (k) i23 y + y 0− 0 y 0 0 v3 = 20 23 23 . (15) npmu = nbr + 1, (17) i65 0 y60 + y56 0− 0 y56 0 v5    −   (k) i67 0 y60 + y67 0 0 0 y67 v7 where nbr denotes the number of branches connected to bus k. A    −   measure of the degree of redundancy is given by: nbu 123456 7 (k) γ = ∑ npmu, (18) k=1 and can be easily incorporated into the IP method and its extensions.  N    N  Placement 1 Similar ideas are used in, e.g., [70, 71]. Note that the degree of redun- dancy has an important effect on the performance of the well-known  N     N Placement 2 robust state estimators, e.g., the least absolute value (LAV) estimator [5] and the Schweppe-type generalized maximum likelihood estimator Fig. 3: The symbol N indicates the bus where the PMU is placed and [72]. The work in [73] modifies the IP method and develops a systematic thus its voltage is directly measured.  indicates that the voltage at that approach to place PMU measurements such that the minimum degree bus can be calculated.  indicates that the voltage at that bus is not of redundancy required by the LAV estimator is achieved. Measurement accessible. redundancy is mission-critical for state estimation and will be discussed in more detail later in this section. In addition to (18), other criteria can be used to prioritize certain locations in earlier stages of a multistage plan or to rank multiple solutions of single-stage placements. For exam- Conversely, v4 cannot be calculated unless Bus 4 is a zero-injection ple, one can prioritize the observability of voltage control areas and/or bus [29]. The system with the measurement Placement 1 is said to be a important tie-lines; see [74]. The multistage minimum PMU placement system with a depth-of-one unobservability because there is at least one problem is approached from a probabilistic viewpoint in [75], where bus in which the voltage is not accessible, linked to two or more buses in the observability of each bus in a network is modeled by a probabil- which the voltage is calculated. Similarly, Placement 2 defines a system ity density function. In other words, as opposed to a deterministic view, with a depth-of-two unobservability. It is stated in [53] that “the crux where each bus in a network is either observable or not, the view in of this approach is that the state [voltage phasor] of buses that are not [75] is that each bus in a network is observable with a given probability. observable can be interpolated from the state of their neighbors rather Accordingly, the problem constraints are modified, and the IP method is accurately,” although the level of accuracy is not discussed. The idea is reformulated as a mixed-integer linear programming problem. Unfortu- to use the degree of unobservability to strategically plan for a phased, or nately, the additional complexity added by the probabilistic constraints is multistage, installation of PMUs. The work in [68] fuses the concept of exchanged by neglecting the zero-injection buses, which is an important degree of unobservability with the IP method and develops a placement deficiency. This deficiency is addressed in [76, 77]. See also [78]. scheme that prioritizes the real-time monitoring of critical buses of the The placement strategies discussed to this point are based on the network. Buses are defined to be critical if they satisfy one or more of notion of topological observability, which is not strictly a sufficient con- the following conditions: dition to solve the state estimation problem (6) [79]. In practice, it is Are high voltage, i.e., 200 kV and more possible to encounter cases in which H is numerically ill-conditioned. • Have many incident branches The work in [80] addresses this issue by developing a simple proce- • Are relevant to rotor angle stability dure [81] to attain numerical observability [see the box Observability for • Are relevant to small-signal stability, static state estimation in transmission grids]. Note that the dimension of • and are selected based on stability assessments performed offline. The H in (1) is m nbu and, in general, m nbu. In particular, m = nbu is the PMU placement problem is formulated to ensure that the voltage phasor size of the minimum× set of linearly independent≥ measured variables for of critical buses is directly or indirectly accessible by at least one PMU, which (1) can be solved. The idea in [80] is to start by building the matrix with weights assigned to prioritize the critical buses. H considering all possible nbu voltage measurements, plus all possi- The weakness of relying solely on the concept of degree of unobserv- ble nbr current measurements, such that, initially, m = nbu + 2nbr > nbu. ability to allocate PMUs in a multistage fashion is that upon completion Then, Algorithm 1 is executed.

pp. 1–18 5 MEASUREMENT CLASSIFICATION Algorithm 1: Incremental reduction of H m number of rows of H; In static state estimation, a measurement is classified as either n ←number of columns of H; critical or redundant. A well-designed measurement system should while← m > n do not contain critical measurements. See, e.g., [88] for more details. 9 condNum0 10 ; A measurement is critical if, after removing it, (1) becomes unob- for k 1 to← m do servable. Errors in critical measurements cannot be detected by bad ← Haux H; data analysis algorithms embedded in the state estimator. Remove← the k-th row of H ; aux A measurement is redundant if, after removing it, (1) remains condNum condition number of Haux; if condNum← < condNum0 then observable. Errors in redundant measurements can always be condNum0 condNum; detected by bad data analysis. ℓ k; ← end ← end Remove the ℓ-th row of H; 2.1.2 Measurement redundancy for static state estimation in m m 1; transmission grids: ← − end To this point, the review is centered around the question of the min- imum set of PMUs that makes (1) observable. For state estimation, however, the observability of the measurement model is not enough. One key task performed by a state estimator is to detect, identify, and correct The rows of H that last upon the completion of Algorithm 1 indi- measurement errors. This task is referred to as bad data analysis [89], and cate the minimum set of independent variables that guarantees numerical it is improved by measurement redundancy. The elimination of critical observability. Note that this method leads to the minimum set of indepen- measurements is of particular interest. This is because the state estimator dent variables, which does not necessarily correspond to the minimum is unable to detect if a critical measurement is a bad measurement. number of PMUs. This important drawback is alleviated by a heuristic A PMU placement strategy that eliminates critical measurements in given in [82], but there is no guarantee that the obtained set of PMUs is the existing RTU measurement system is developed in [90] and extended minimum. This drawback is effectively overcome in [83] by combining in [91]. The work in [90, 91] develops on and expands the applicability the IP method with the original ideas in [80]. The work in [55] relies on of the IP method. It starts with a clever observation that from a topolog- similar ideas, i.e.: i) build H for an initial PMU placement; ii) remove ical standpoint, the system observability is independent of the numerical value of the parameters, yk ,yk , and the algebraic state vector, x. This a PMU and update H; iii) check if the updated H is of full rank. The { 0 ℓ} solution is obtained iteratively. allows for series admittances to be set equal to j1.0 per unit and volt- age magnitudes to be set equal to 1.0 per unit. Shunt admittances are Consideration of PMUs with limited measurement channels: neglected. We stress that (topological) observability analysis does not depend on the actual state of the system or the branch parameters, mak- To this point, it is assumed that a PMU installed on bus k mea- ing it possible to use these simplifications without loss of generality [5]. sures the voltage phasor at bus k as well as the current phasor in all By plugging values into: the branches that are incident to bus k. This is possible because of the jθk jθℓ jθk jθℓ capability of the commercially available PMUs that typically have more ikℓ = ykℓ vk e vℓ e = j1.0 e e , (24) than 20 measurement channels; however, this capability could be limited, | | −| | − − thereby influencing the problem solution. To this end, [84, 85] use a strict and after some algebraic manipulations, one obtains a linear regression assumption that PMUs are supposedly provided with two measurement model between currents and voltage phase angles: channels—one for the voltage signal and the other for the current sig- Re(ik ) θk θ , (25) ℓ ≈ − ℓ nal. Moreover, though previous work supposes that PMUs are installed that is valid for sufficiently small θ and θ , such that: on the network buses, here PMUs are supposedly installed on the net- k ℓ . . . work branches. Accordingly, the optimization problem in (8)–(11) is . .. . reformulated as follows: . . θk 1 0 θk nbr   =  ··· ···   = Hpmu θ. (26) Re(ikℓ) 1 1 θℓ · min c′ u′ , ··· − ··· ∑ k · k  .   .  .  k=1 (19)  .   ..  .   .    .  s.t. g(u′) 1,      ≥ Note that the algebraic manipulations in (24)–(26) are necessary to where nbr is the number of branches in the network, ck′ is the cost of merge the measurement model of the RTU measurement system, which installing a PMU on branch k: is nonlinear, with the measurement model of the PMU measurement 1 if a PMU is installed on branch k, system, which is linear. Next, instead of (1), consider the measurement u = (20) model: k′ 0 otherwise,  zP = HPθ θ + e, (27) nbr where zP contains measurements of real power flows and injections, and gk(u′)= Bk u′ 1, (21) ∑ ℓ · ℓ ≥ HPθ is the Pθ sub-matrix of the Jacobian matrix obtained from the DC ℓ=1 power flow equations. See also [92, 93]. By augmenting (27) with (25): 1 if branch ℓ is incident to bus k, H B = (22) Measurement matrix = Pθ , (28) kℓ 0 otherwise, Hpmu  and it is possible to consider mixed measurementh i systems of RTUs and Bkℓ denotes the kℓ-th element of the bus-to-branch connectivity and PMUs altogether. The measurement matrix in (28) is used along matrix B [see the box Admittance matrix and connectivity matrices]. with the IP method to place PMUs and eliminate critical measure- For example, for the network in Fig. 2: ments simultaneously. See [91] for more details and [94, 95] for an 10000000 insightful discussion. The computational efficiency of the method in [91] 11110000 can be improved by resorting to the lower-upper decomposition of the measurement matrix along with sparsity techniques [93].  01001100  B = 00001011 . (23) In addition to eliminating critical measurements, one must consider leverage measurements [see the box Leverage measurement]. This is  00000010   00100100  because leverage measurements have an undue effect on most power sys-   tem state estimators, with a few exceptions [72, 96, 97]. Note that most  00010001    energy management systems available commercially employ a WLS As in [24], by modifying the constraints g(u′), it is possible to con- state estimator, which is vulnerable to leverage measurements. Indeed, sider zero-injection buses, eliminate critical measurements [see the box the vulnerability of the WLS estimator to bad data is known since the Measurement classification], and retain observability under a set of con- seminal work of Schweppe and Handschin [89, 98]. Moreover, as pin- tingencies and single PMU losses [84, 85]. Interestingly, a set of 1,291 pointed in [96], locations susceptible to create leverage measurements PMUs resulted from solving (19) for a large utility system with 2,285 “have to be provided with enough measurements in order to increase buses. Note that 1,291 > nbu/2, which is, in principle, the worst-case their local redundancy. Indeed, they tend to be isolated in the factor scenario [16]. This result provides evidence that the minimum number space and weakly coupled with the surrounding measurements.” Thus, of PMUs required to attain system observability might be considerably a cluster of measurements around these locations is recommended to augmented by considering practical factors. This is clearly the case of bound their influence in the estimation process; otherwise, the algebraic devices with limited measurement channels. A set of devices with more states (voltage phasors) close to these locations are estimated with large measurement channels tends to yield a larger spanning measurement variances. Unfortunately, though the formulation in [90, 91] eliminates tree. For other works that consider the case of limited measurement critical measurements, it does not consider leverage measurements. The channels, see [86, 87]. problem of measurement redundancy is only partially solved in [90, 91].

pp. 1–18 6 LEVERAGE MEASUREMENT supposes that the system is observable through the legacy measurement system, but it brings in a new perspective—focuses on placing PMUs to reduce the variance of estimated quantities [see the box Residual sen- Recall the measurement model (1), which is given by z = Hx + e. T sitivity analysis]. By reducing the variance of estimated quantities, one Let zi be the i-th element of z, and hi be the i-th row of H. All improves the accuracy of the state estimation process. T It is cumbersome to consider variance reduction in formulations using the row vectors hi , i = 1,...,m , lie in the so-called factor space of regression. If there is an{ outlier in} hT, then the corresponding mea- the notion of topological observability. This is the case, for example, of i the previously described IP method and its extensions. An error covari- surement zi will have an undue influence on the state estimate x. ance matrix exogenous to the original formulation would have to be built See (6). In this case, zi is a leverage measurement. For more details, the reader is referred to [5, 96]. Note that this definition of a lever- and assessed iteratively. Conversely, the notion of numerical observabil- b ity lends itself better by intrinsically considering the covariance matrix age measurement is based on the notion of factor space, and it is within the formulation. In this context, the idea of using the covariance anchored on statistics theory. The notion of leverage measurements matrix to place measurements can be traced back to the seminal work of was introduced to the power system community by Mili, Rouseeuw, Fred C. Schweppe and colleagues; see, e.g., [106] (page 124, Discussion and colleagues [96]. So-called by statisticians as influential data of Theory), or [98] (page 980, Meter Configuration). These works use points, it was Rouseeuw and colleagues [99, 100] who coined the the numerical observability-based problem formulation. term leverage point. We should mention, however, that the numerical From the standpoint of an optimal experimental design, a reasonable effect of leverage measurements on power system state estimation 1 goal during the design phase of an experiment is to minimize G− in was noticed earlier by Monticelli [79]. The following conditions are 1 known to create leverage measurements in power systems: some way. Note that G− has a direct impact on the state estimate x; 1 see (6). There are many different ways in which G− might be made An injection measurement placed at a bus that is incident to a minimal. Three optimality criteria considered in [105] are: large• number of branches b 1 An injection measurement placed at a bus that is incident to A-optimality: minimize the trace of G− . • • 1 branches of very different impedance values D-optimality: minimize the determinant of G− . Flow measurements along branches whose impedances are very • E-optimality: minimize the maximum eigenvalue of G 1. different• from those of the other branches in the system • − Using a very large weight for a specific measurement. The work in [105] is extended in [107, 108], wherein the mutual infor- • mation is the adopted criterion. Both works use a greedy algorithm to solve the problem numerically. The idea of placing PMUs to improve the state estimation performance from an optimal experimental design standpoint is also pursued in [109–111], wherein the problem is solved The consideration of leverage measurements is a gap in the literature after relaxing it to a convex semidefinite program. In particular, [110] and an opportunity for future developments. This is further discussed in formulates a multicriteria framework that concurrently seeks to: Section 4. Increase measurement redundancy Consideration of topology changes and topology errors: • Eliminate critical measurements • Maintain observability in case of single branch outages In addition to critical measurements, topology changes are another • factor that might render a measurement system with minimally placed Attain E-optimality • Improve the convergence of the Gauss-Newton algorithm. PMUs ineffective. Note that topology changes occur ever so often, • and they are driven by, e.g., scheduled maintenance of equipment, The latter is noteworthy because [110] is the first attempt to con- seasonal trends in electric power consumption, and unforeseen equip- sider the numerical convergence of the state estimation solution process ment/systems outages. in the PMU placement problem. Note, however, that the iteratively Additionally, the measurement matrix, H, needs to be reevaluated reweighted least-squares algorithm [112] is preferred over the standard after any topology change. This is accomplished by topology processing Gauss-Newton algorithm because of superior numerical stability. See algorithms embedded into the static state estimators. These algorithms also [113], wherein the optimal design considers the fact that PMU are responsible for building the network connectivity model—also measurements of phase angles (for both voltages and currents) are not referred to as a bus-branch model—based on the status of switchgear perfectly synchronized; and, [114], wherein the robust LAV estimator in the field; however, it is not uncommon for the equipment statuses to is considered in addition to the classic WLS estimator. The works in be reported incorrectly, e.g., because of communications issues. This is [105, 107–111, 113, 114] also offer interesting theoretical developments, an issue because topology errors yield biased estimates and might cause particularly in the areas of signal processing and optimization. the state estimator to diverge. In power systems, however, the standard deviations of the noise of the Therefore, a reliable measurement system should guarantee observ- metering devices and their associated communications channels, includ- ability in case of topology changes and in case of erroneous switchgear ing the PMUs, are estimated with large uncertainties [115, 116]. This status [101]. By relying on the original ideas in [16], [30] addresses N- fact adds to the level of uncertainty of any approach that, directly or 1 contingencies, and [37] addresses single branch outages. In general, indirectly, uses G 1. Moreover, to each state x, there is a corresponding to ensure topological observability in case of contingencies, additional − constraints can be imposed on the IP formulation as follows: nbu c c gk(u)= ∑ Akℓ uℓ 1, (29) ℓ=1 · ≥ RESIDUAL SENSITIVITY ANALYSIS where the constraint gc enforces the topological observability of bus k k c in case of a contingency c, e.g., the loss of a branch or a PMU; Akℓ Define z := Hx. Then, using (6): k 1 denotes the ℓ-th element of the bus-to-bus connectivity matrix of the T 1 − T 1 system under contingency; and uℓ is defined as in (9). Also, some work z = H H R− H H R− z = Sz, (30) has addressed the case of single outages by enforcing all buses to be b b observed at least twice by PMUs—to do so, one simply sets the right- where S is referred to as the hat matrix. Now, define the residual hand side of (10) to be 2 instead of 1. vector r := zb z. Thus: ≥ ≥ − The work in [102] builds on [80] and addresses single measurement r = z Sz = (I S)z = Wz = W (Hx + e) losses and single branch outages. These strategies remedy specific cases − b − of topology changes but do not address the case of erroneous switchgear = W Hx + W e = (I S)Hx + W e status. Conversely, [103] expands on (24)–(28) and develops a PMU − 1 placement strategy that ensures that any single branch topology error T 1 − T 1 = Hx H H R− H H R− Hx + W e is detectable. As a by-product of the developed strategy, it is guaran- − teed that the system will remain observable in case of any single branch     contingencies. The approach in [103] accounts for a systematic way to = W e, (31) address the problem of topology changes and is overall superior to the others. Note that the approach in [103] does not account for leverage where W is referred to as the residual sensitivity matrix. Now, under measurements either. the assumption that the measurement errors are Gaussian: Other ideas with limited impact appear in the literature. An example E (r)= E (W e)= W E (e)= 0, (32) is considering controlled islanding. Provided the system has a suffi- T T T T T T cient degree of observability, including controlled islanding [104] in the E (rr )= E (W ee W )= W E (ee )W = WRW . (33) measurement placement problem is unnecessary. T T T It can be verified that RW = RW . By plugging this into 2.1.3 State estimation accuracy under ideal conditions: (33) and using the fact that R is a diagonal matrix: T
There is prolific work on the minimum placement of PMUs for sys- E (rr )= WWR = WR, (34) tem observability. Some also consider the legacy measurement system based on RTUs. For example, [90, 91] develops a systematic approach where WW = W because W is an idempotent matrix; hence, the to eliminate critical measurements in the legacy measurement system by residual sensitivity is related to the error covariance matrix, R. strategically allocating PMUs. Following [90, 91], the work in [105] also

pp. 1–18 7 Analyze planning model to eliminate Based on Identify Solve mul- main cost Solve basic Virtual buses Minimal observability solution tistage problem components • Radial buses • Budget problem • Zero-injection buses • Project timeline • •

Cost of For minimal observability considering Considering Define Define Measurement devices Main cost components • Instrument transformers candidate • strategy Main cost components • Preselection of candidate buses • Preselection of candidate buses Communications locations • Zero-injection buses • • Data storage • Zero-injection buses • • Existing infrastructure • Critical measurements • Leverage measurements • Loss of measurements devices • Loss of communications link • Branch outages • Critical buses priority • Estimation accuracy • Fig. 4: Steps and summary of the most important factors to consider in the measurement placement problem. The focus is on only static state estimation in transmission grids.

1 1 matrix G− . In other words, G− varies depending on the system operat- grids, it is common to encounter buses that do not exist physically or 1 ing condition. The use of a single matrix G− is, therefore, discouraged are not in practical locations to install PMUs. The elimination of these because it yields a plan that focuses on a single operating scenario. This buses, though it does not affect the final solution, significantly reduces drawback can nonetheless be circumvented by using multiple operating the effort to find it [125]. Radial buses are another important consid- conditions sampled, e.g., through Monte Carlo simulations. eration. For example, in the network in Fig. 2, buses 1,5 are radial buses. Consider Bus 1. To make Bus 1 observable, there{ are} only two Consideration of additional costs: candidate buses to pick from: 1,2 . Installing a PMU on Bus 1 leads The discussion until now focuses on the cost of placing PMUs subject to buses 1,2 being observable,{ whereas} installing a PMU on Bus 2 to constraints, which are specifically tailored to guarantee a prede- leads to buses{ } 1,2,3,6,7 being observable. The second option is bet- fined goal, e.g., system observability. But a study conducted by the ter. This observation{ can always} be leveraged to eliminate radial buses U.S. Department of Energy reveals that the cost of PMUs represents from the candidate set. Another important observation relates to zero- approximately 5% of the total investment [17]. The communications injection buses. As discussed in previous sections, the consideration of infrastructure is unquestionably an important layer between the mea- zero-injection buses leads to a reduction in the set of constraints; thus, in surement devices and the end-user application [117]. This aspect is general, it is recommended to not include zero-injection buses in the set overlooked in most of the previously discussed work. A few exceptions of candidate buses. include [53, 102], in which this aspect is superficially touched upon. Before moving on to the next section, we present a summary of impor- The work in [40] accounts for as an attempt to expand on that front by tant factors. See Fig. 4. These factors must be addressed when the focus co-optimizing the cost of PMUs and the cost of building the communi- of the measurement placement problem is on static state estimation. cations infrastructure. The cost of the communications infrastructure is explicitly formulated as follows: 2.2 Dynamic state estimation in transmission grids nlink min ∑ clink,k λk, (35) 2.2.1 Observability for dynamic state estimation in transmission k=1 · grids: where nlink is the number of communications links necessary to form a connected graph with the PMUs; c is the cost per unit of length Dynamic state estimation has gained momentum in the power system link,k community. Static and dynamic state estimation are significantly differ- of the communications link k; and λk is the length of the communica- tions link k. The IP method is augmented with (35), and the solution is ent in many aspects, such as modeling and assumptions, the range of obtained through a multi-objective genetic algorithm. Further costs are applications, and the requirements imposed on the measurement systems considered in [48, 51, 118, 119]. In particular, [48] reports on the experi- and communications networks. See, e.g., [126–129]. Here, the important ence obtained from the expansion of the wide-area measurement system difference between the two is in the measurement model (1), specifi- of the Danish transmission grid. Although most extensions of the IP cally, in the vector x. For static state estimation, x denotes the algebraic method overlook the dissimilarity in the cost of installing PMUs on dif- state vector, which contains the voltage magnitude in all buses and the ferent buses and focus on the set of constraints in the pursuit of specific voltage phase angle in all but the reference bus. As for dynamic state enhancements, [48] focuses on the overall cost and reveals a set of impor- estimation, x denotes the dynamic state vector or simply “state vector” tant hidden factors that must be included in the optimization model. The (the latter is more common outside the power system community). In discussions related to instrument transformers are particularly notewor- this case, x contains the variables associated with generators and their thy. The cost of a set of three potential transformers (PTs) plus three controllers, for example, the rotor speed of synchronous generators and current transformers (CTs)—one PT and CT per phase in a three-phase the pitch angle of wind turbines. Because x is different, H is different. system—is approximately six times the cost of a PMU. This does not More importantly, for dynamic state estimation, H is time-varying. The include the cost of the structural foundation that is necessary to install notion of observability is thus more involved. In the case of dynamic the instrument transformers in the substation. Nonetheless, multichannel state estimation, as articulated in [130], “higher (lower) values of the PMUs require three CTs per current measurement channel. Also, there smallest singular value of the observability matrix indicates stronger are additional hidden costs. For instance, it might be necessary to curtail (weaker) observability for a given measurement set. Since observabil- generation or load for several hours to install the measurement system, ity is a local property, the smallest singular value of the observability from which shutdown costs will incur. It is interesting that though critical matrix will change along the trajectory of x.” By contrast, the notion of facilities such as large generation plants are taken as preferred locations observability in static state estimation is binary—that is, either the sys- to install PMUs, they might incur the highest shutdown costs. Also, the tem is observable or not observable—and not time-varying. Therefore, installation of instrument transformers involves a significant amount of the problem of observability for dynamic state estimation is more chal- man-hours for engineering, cabling, etc. Fortunately, it is straightforward lenging. And so is the problem of measurement placement that seeks to to include these costs in the formulation of the IP method. Reference [48] attain observability. also provides a cost baseline that might be useful for initial projections. The work in [131] is the first to formulate a PMU placement problem For costs associated with substation infrastructure, see [120, 121]. For an that considers dynamic state estimation. The classical synchronous gen- informative discussion on the communications infrastructure, see [122]. erator model [132] is adopted, and the strategy is as follows. In the first See also [123, 124]. stage, the IP method is executed to obtain the minimum set of PMUs that yields (1) as topologically observable for static state estimation; this Preselection of the set of candidate buses: typically leads to multiple solutions. Then, in the second stage, the solu- tions are ranked using a criterion that relates to dynamic state estimation. In practice, before attempting to solve a heuristic or an optimization The criterion is based on the asymptotic error covariance matrix of the model, it is necessary to build a set of candidate locations to place mea- Kalman filter, given by: surements. In this process, some buses might be selected for mandatory T PMU installations, whereas others can be eliminated, thereby reducing lim E (x x)(x x) , (36) k ∞ − − the effort to search for an optimal solution. The latter is very important → because the problem of finding the minimum PMU placement for sys- where x and x denote, respectively,h the truei and estimated dynamic tem observability is NP-complete [33]. In the models of electric power state vector. Note that although theb Gauss-Newtonb or the iteratively b

pp. 1–18 8 d reweighted least-squares algorithms are used in the solution process of The coherency matrix, C— that is, I Z( ) = C . the static state estimation, dynamic state estimation relies on Kalman fil- • C || || tering. The work in [133] proposes the use of lower and upper bounds The choice in [148] is the Gramian norm, which is defined as: instead of the asymptotic error covariance matrix. The work in [134] T 1/nc does not consider static state estimation and focuses exclusively on (d) (d) (d) dynamic state estimation. It formulates an optimization model to max- I ZC = det I ZC I ZC . (40) imize the determinant of the empirical observability Gramian, which   h  i h  i is calculated for a set of operating points to quantify the degree of The kℓ-th element of E is given by: observability of a given PMU placement. See also [135]. Dynamic state estimation will play a critical role in power systems ∞ zkℓ(ω) control and protection [136]. Despite the availability of preliminary Ekℓ = Eℓk = log dω, (41) results, the problem of measurement placement for dynamic state esti- Z0 zkk(ω)zℓℓ(ω) ! mation remains open. Accordingly, measurement placement for dynamic state estimation is another opportunity for future developments. This is where zkℓ(ω) is the power spectralp density for k = ℓ, and the cross further discussed in Section 4. spectral density otherwise. The kℓ-th element of C is given by:

T 2.3 Stability assessment and control in transmission grids 1 2 2 Real-time stability [137, 138] assessment and control represent another Ckℓ = Cℓk = [θk(t) θℓ(t)] +[ fk(t) fℓ(t)] dt , (42) s T Z1 − − major application of PMUs. This is discussed next.   where T denotes the sampling period, and fk denotes the frequency at 2.3.1 Voltage control in transmission grids : bus k. The optimization model is defined as: The work in [28] studies the effect of PMU location on the secondary (d) voltage control of transmission grids; it argues that the minimum set of min I ZC , PMUs that makes (1) observable is sufficient for systematically identify- (43)   ing pilot points. Note that, following [28], a “pilot point is a voltage at a s.t. np nc, load bus which is measured in real-time and used for control action.” ≤ The pilot point must be representative of all the voltages within the where np denotes the number of placed PMUs. See also [149]. region where it is located. The adoption of the notion of pilot points, The consideration of other notions of stability—particularly fre- however, might be obsolete in the context of the ongoing moderniza- quency, resonance, and converter-driven stability [138]—is a gap in the tion of electric power grids. This is because of the sustained growth in literature and an opportunity for future developments. This is further the number of flexible AC transmission system devices deployed world- discussed in Section 4. wide. Also, an investigation of the effect of incremental PMU placement on decision tree-based online voltage security monitoring [139] reveals 2.4 Topology change detection in transmission grids an overall improvement in voltage security misclassification rates. There have been some attempts to use PMU measurements to detect line The work in [140] combines the multistage ILP method with a outages directly, without requiring state estimation. Most available meth- measure of the system dynamic performance. The idea is to perform ods assume that if a line outage occurs, then the voltage phase angles time-domain simulations and rank the system buses according to their change significantly in response to the change in topology. The work dynamic vulnerability. The buses are divided into generator buses and in [150] makes the additional assumption that the power injections of load buses. The vulnerability of the generator buses is calculated using the network remain the same within a few seconds after a line outage the individual machine energy functions [141]. The vulnerability of the occurs. Also, it employs the DC power flow model to do offline simula- load buses is calculated using the concept of proximity to voltage col- tions and collect signatures of the system’s voltage phase angle responses lapse [142]. Based on the obtained bus ranking, the most vulnerable to single-line outages. Based on these assumptions, the optimization buses are made observable at the earlier stages of the multistage place- objective is established as one of maximizing the minimum distance ment. A similar idea is pursued in [143], wherein buses are ranked based among the voltage phase angle signatures of the outages. The prob- on their correspondence to the largest Lyapunov exponent of the network lem is formulated as an integer program and solved by using a greedy [144]. The idea is that buses with a strong correspondence to the largest algorithm. Interestingly, for the IEEE 30-bus system, if 10 PMUs are to Lyapunov exponent are more significant from the system stability stand- be installed to detect line outages, the optimal locations are found to be point, and they should be made observable with higher priority. See also at buses 1,5,8,9,14,21,22,24,26,29 . See also [151]. Now, based on [145]. The drawback of these approaches lies in their dependency on an the exhaustive{ search in [54], the minimum} number of PMUs to make explicit energy function, which is challenging to obtain. the IEEE 30-bus system observable under normal operating conditions, considering zero-injection buses, is equal to 7. If single branch outages 2.3.2 Rotor angle stability assessment in transmission grids: are considered, then this number increases to 10. The optimal locations The work in [146] focuses on (small-signal) rotor angle stability. of PMUs obtained in [54] are as follows. Considering: Consider the similarity transformation: 1 Normal operating conditions: 1,2,10,12,15, 19 or 20 ,27 w = U − x, (37) • Single branch outages 2,3,5,{10,12,15,17,19{,24,27 .} } where w denotes the vector of the modal variables and U the matrix • { } containing the right eigenvectors of the linearized system model. By Note that the optimal locations in [150] and [54] are hardly compa- plugging (37) into (1), one obtains: rable. This provides a clear illustration that: i) the optimal solution for z = HUw + e, (38) a particular application might not be the overall best approach, and ii) a formal relationship between the measurements and modal variables. considering different strategies for optimization separately might lead to The key idea in [146] is to use the matrix product HU to find a set conflicting solutions; therefore, such multiple considerations will be nec- of measurements from which all inter-area modes are observable. This essary to develop a coordinated placement strategy that is cost-effective method has two important drawbacks: and applicable to many use cases. In [152], a logistic regression-based Its performance on highly meshed networks might be poor. Note that method is employed to identify the most influential buses for outage electric• power transmission grids are typically highly meshed. detection. See also [153]. It relies on information obtained from the eigendecomposition of the 2.5 linearized• system model. This information can vary significantly under Fault detection in transmission lines different operating scenarios. The work in [154] is the first to propose a PMU placement scheme for The work in [147] develops a simplistic approach to place PMUs fault location. The adopted heuristic is simple, and the algorithm has on tie-lies with the objective of monitoring the internal voltage of syn- basically two rules: i) place PMUs on the two buses with the largest num- chronous generators. It focuses on the (transient) rotor angle stability ber of connected branches; ii) such that between two PMU buses there of reduced two-area power systems and, because of that, has limited is a bus with no PMU. The solution obtained with this simple scheme applicability. A more elaborate strategy is available in [148], which is not unique, and the second rule might lead to more PMUs than the minimum needed for fault location; thus, the solution always needs to starts by considering the set of all network buses, B, card(B)= nbu; a set of candidate buses, C, card C n ; and a set of credible distur- be refined. More importantly, the number of PMUs required for fault ( )= c location is much larger than the minimum number of PMUs required bances, D, card(D)= nd . Then, for a given disturbance d D, define the measurement matrix: ∈ for observability in static state estimation. For example, in the IEEE 14- (d) bus system, 3 PMUs are required for observability [16] and 8 PMUs are ZC =[z1(t) z2(t) ... znc (t)], (39) required for fault location [154]. See also [155–157]. where zk(t) is a vector containing m pseudo-measurements of the bus variable associated with the k-th element of C. These pseudo- 2.6 Power quality monitoring in transmission grids measurements are generated through the numerical simulation of each The first work to provide insight into how to place measurements for disturbance d D. The strategy is to select the set of PMUs, S C B, power quality monitoring is [158], which describes a reverse power flow ∈ ⊆ ⊆ that maximizes the information content of Z(d), denoted by I Z(d) , procedure to identify the source of harmonics in electric power grids. C C Notice that power quality monitoring requires a specific type of meter, which is quantified by some norm (typically the ℓ2-norm) of:   referred to as harmonic meter or power quality meter. The procedure relies on a linear relation between the Fourier transforms of bus voltages, The entropy matrix, E— that is, I Z(d) = E ; or • C || || v(ω), and bus injection currents, i(ω), as follows:   pp. 1–18 9 Another aspect to consider is the system infrastructure: existing i(ω)= Y (ω)v(ω), (44) instrument transformers, availability of communications systems, space for expansion, etc. The nodes that represent a transmission grid always where Y (ω) is the bus admittance matrix. Let io(ω), vo(ω) denote a reside within a high-voltage substation, in which adequate infrastructure vector of observed or measured quantities, and let iu(ω), vu(ω) denote is often available. On the other hand, many nodes that represent a dis- a vector of unobserved quantities, such that: tribution grid are located in a section of an overhead (or underground) iu(ω) vu(ω) cable crossing cities with no or minimal infrastructure. This imposes i(ω)= , v(ω)= (45) additional constraints on the candidate locations for measurement place- io(ω) vo(ω)     ment in distribution grids. These and other aspects are discussed next. It follows that: As in the previous section on transmission, in this section, we elaborate i Y Y v on the optimal measurement placement problem as well as on challenges u = uu uo u , (46) io You Yoo vo and solutions for distribution systems for several different applications, where (ω) is omittedh for simplicityi h of notation.ih In [158],i the unobserved starting with static state estimation. quantities are estimated in the least-squares sense: 3.1 Static state estimation in distribution grids 1 T T − vu = You YouYou (io Yoovo), (47) 3.1.1 Observability for static state estimation in distribution − grids:   iu = Yuuvu + Yuovo, (48) Table 3 shows that, irrespective of the measurement system, distribu- b tion grids are not observable. The following remarks are in order. and it is suggested to place measurements to reduce the condition num- b Remark 1 (Observability in electric power transmission grids). To this ber of the matrix You. Theb rationale of this strategy is provided, but no point, two definitions of observability are given: numerical observability guidelines on how to choose the measurement locations are offered. Yet, existing linear algebra-based algorithms can pinpoint the best variables and topological observability. See the box Observability for static state (measurement locations) to choose from to improve the condition num- estimation in transmission grids in Subsection 2.1. We refer to these ber of a given matrix. See, e.g., [159]. Conversely, in [160], the problem definitions as strong notions of observability; they are widely used in is approached in the following manner. For a given set of candidate transmission grids. measurement locations and a predefined number of measurements to Remark 2 (Observability in electric power distribution grids). Static be placed, find a measurement configuration that minimizes the error state estimation of distribution grids relies heavily on the use of pseudo- iu iu. This approach follows the idea of using the covariance matrix measurements [166, 167], which are obtained from historical load data. to− place measurements [98, 106]. The evaluation of all possible mea- Without the use of pseudo-measurements, distribution grids are not surement configurations through a complete enumeration method is a observable according to the strong notions of observability. bformidable task. For example, to place 5 measurements in a system of 100 Definition 1. Weak numerical observability is defined as the ability of 100 buses would require the evaluation of 5 = 75,287,520 possible the linear model (1) to be solved for a state estimate x, provided that the combinations; hence, a sequential solution process is adopted in [160] measurement vector z is augmented with pseudo-measurements. under the assumption that the best k + 1 measurement
locations contain the best k locations for all k. It turns out that this sequential scheme is not It is clear from the previous discussion that observabilityb is yet guaranteed to yield an optimal solution, as numerically demonstrated in a challenging concept to be applied in distribution grids. See [168]. [160]. To circumvent this issue, a genetic algorithm is proposed in [161]. For this reason, the placement of various measurement technologies— It is numerically demonstrated on small-scale test systems that the solu- particularly when they are simultaneously considered—remains of high tion achieved using a genetic algorithm is optimal and hence superior to interest, and an optimal mix is contemplated in what follows. Note that the sequential solution. this increases the complexity of the problem; whereas in Section 2 the The work in [162] extends the methodology developed in [80] to the goal was to find the number and location of measurements, in this section specific application of power quality monitoring. Refer to Subsection the goal is to find the number, location, and type of measurements. Table 2.1, Algorithm 1. See also [163, 164]. 4 summarizes the types of measurements considered in this subsection. Also, in transmission grids, the algebraic state variables are defined as 3 Measurement placement in distribution grids the voltage phasors (magnitude and phase angle) at each bus.‡ Accord- In the U.S. electric power grid infrastructure, “there are approximately ingly, the static state estimation algorithms are designed to estimate the four times more low-voltage distribution substations than there are high- voltage phasors based on a set of measurements. Keep in mind that the voltage substations” [165]; and each low-voltage distribution substation choice of algebraic state variables is not unique. In distribution grids, minimally houses one feeder that delivers power to a neighborhood— two definitions of algebraic state variables are commonly used: this implies that if represented by graphs, distribution grids certainly Voltage phasors at each bus have at least 10 times as many nodes as transmission grids. This is a • Current phasors at each branch. key difference between transmission and distribution grids, and it has • It is beyond the scope of this paper to discuss the advantages and an important effect on measurement placement. Keep in mind, however, disadvantages of these two options; see, e.g., [169, 170]. Of importance that this difference in the number of nodes does not necessarily translate here, however, is the following. Given that the measurement placement linearly into the technical requirements for (and cost of) measurement has a direct impact on the performance of the static state estimator [171, placement in distribution grids. It does, however, support the argument 172], the choice of algebraic state variables will affect the formulation that a careful measurement placement is of paramount importance. Table of the measurement placement problem. 3 summarizes additional characteristics that distinguish distribution from Having motivated the measurement placement problem and discussed transmission grids. its particularities to distribution grids, let us proceed to existing methods. Minimum number of measured quantities for weak observability: Table 3 Distribution vs. transmission grids: Comparison of characteristics A heuristic approach tailored to radial feeders is proposed in [173, 174]. A set of rules developed based on empirical observations is Characteristic Distribution grids Transmission grids proposed to determine the number, location, and type of meters, as Topology Radial to slightly meshed Highly meshed follows: Phase unbalance / Significant / each phase Negligible to moderate / circuit analysis is analyzed individually analysis of one phase Place a power meter at the substation. often suffices • Place current meters on all main switch and fuse locations that need Measurement system / RTU / unobservable, RTU / observable, to• be monitored. observability Micro-PMU / unobservable PMU / unobservable† Place current meters along the feeder such that the total load in the Renewable generation / Dispersed / high / very high Lumped at high-voltage zones• defined by the meters are similar in magnitude. operational†† uncertainty / substations / moderate / Place current meters on all normally open tie switches used for feeder planning‡‡ uncertainty high to very high switching.• Measurements of voltage magnitude at both ends of these tie Load demand / Lumped at distribution Lumped at high-voltage switches are desirable. operational uncertainty / transformers / high / substations / moderate / planning uncertainty very high high to very high The number of meters placed on the system if the previous rules are used alone might be prohibitively large. The authors circumvent this problem by adapting the method as follows [197]. First, the notion of interesting or influential quantities is defined as any variable that can be expressed in terms of the algebraic state variables and is needed by monitoring and control applications, referred to as distribution automa- tion functions. Then, the set of meters obtained with the previous rules †A few transmission grids in the United States are fully observable by PMUs [129]. ††Real time to a few hours ahead. ‡In certain cases, in addition to voltage phasors, the transformer taps and ‡‡Day ahead to weekly to yearly. the firing angle of converters are also defined as algebraic state variables.

pp. 1–18 10 Table 4 Measurement placement methods for static state estimation of distribution grids (ordered by year of publication) Meter (Measurand) [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [168] [189] [190] [191] [192] [193] [194] [195] [196] Pseudo (kwatt, kvar) XXXXXXXXXXXXXXXXXXXXXXXX Current (amp) X X X X X X X Voltage (volt) XX XXXXXXXXX XXX XX X Power (kwatt, kvar) X X X X X X X X X X X X X X X X X (kwatt only) PMU XXX XX Micro-PMU X Smart meter X X X X Algebraic state variables Voltage phasors XXXXXXXXXXXXXXXXXXXXX Current phasors X X X is ranked based on their impact on the variance of the estimated interest- Build sets of true values for N operating conditions by solving a power ing quantities. The ranking will indicate the order in which the meters flow• algorithm for different random loading scenarios. are to be eliminated, if needed. This method represents the first reported For each operating condition, build sets of synthetic measurements by attempt to place measurements in a principled fashion to aid in the static adding• random noise to the power flow variables. state estimation of distribution grids. Along similar lines, the work in [175] relies on empirical observations to develop a rule-based heuris- Next, assume that measurement devices are installed at the substa- tic approach aimed at reducing the variance of the estimated voltage tion and DG buses. Accordingly, synthetic measurements corresponding to these locations are used in an attempt to solve (49). If a solution is magnitudes on those nodes that are not measured. As opposed to the C deterministic approach taken in [173, 174], uncertainties in the feeder found, no additional measurements are required. Else, define a set of loads, network parameters, and the calculated feeder node voltages are candidate locations for measurement placement. Also, define: considered in [175]; more specifically, the uncertainties are characterized 1 N by ranges of values—i.e., by intervals with confidence levels represented ε = ∑ Jℓ. (50) as fuzzy numbers—and the measurement placement scheme is assessed N ℓ=1 on all plausible system operating conditions. The rule-based approaches Then, beginning from the initial configuration with measurements at in [173–175] (and their extensions) are simple but need to be adapted on the substation and DG locations: a case-by-case basis, thereby requiring specific knowledge of the system. See also [198]. 1. Evaluate ε for all candidate locations in C. An important characteristic of modern distribution grids is the pres- 2. Place a measurement at the candidate location that yields min(ε). ence of distributed generators (DGs), which are not considered in the 3. Remove the corresponding candidate location from C. previous work. An endeavor to fulfill this gap is reported in [176], where 4. If min(ε) < preset value and the constraints in (49) are not violated, the case study contemplates wind power plants connected to the 11- stop; else, go to 1. ¶ kV section of the U.K. generic distribution system (UKGDS); and in The procedure is illustrated in Fig. 5, where C = a,b,...,z . Starting [177, 178], where the case study contemplates portions of an Italian dis- { } tribution grid to which wind, gas, and cogeneration power plants are from the initial configuration, ε is evaluated for all candidate locations, connected. Following [175], these works rely on the variance of volt- and c yields min(ε). An additional measurement is placed at this loca- age [176] or current [177, 178] magnitudes estimated for those nodes tion, but the stopping criteria are not met; thus, ε is reevaluated for each that are not measured as a metric to assess the quality of the obtained of the remaining candidate locations, and z yields min(ε). This process state estimation results. See also [168, 181, 192]. is repeated again before the stopping criteria are met. At the end, three The method in [176] is heuristic and can be summarized as follows: additional measurements are placed at locations b,c,z . It is reported in [178] that approximately 29,000 combinations{ are evalua} ted by this 1. Define an initial set of buses, S, where voltage meters are installed. method before a suboptimal solution to (49) is achieved for a distribu- 2. Solve a power flow algorithm for the peak load. tion grid of 51 nodes. For the same grid, an optimal solution to (49) using 3. Solve a power flow algorithm for a random, 20% of nominal load to a complete enumeration method would require the evaluation of combi- ± consider uncertainties in load. Note that one can modify this algorithm nations in the order of 1013. The algorithm developed in [177, 178] is to consider variations in net load, that is, load minus variable distributed further exploited in [183] with an extended error covariance matrix that generation. accounts for network model parameter uncertainties. card(S) peak 2 4. Calculate the difference: e = ∑ V V rand . An extension of the previous methods is developed in [179, 180] in k=1 k − k which the variance of voltage magnitudes and phase angles, estimated 5. If e < preset threshold, store the power flow case obtained in 2. on those nodes that are not measured, is used as a metric to evaluate the 6. Repeat 2–4 until N (preset number of) power flow cases are obtained. obtained results. The authors start by defining: 2 ∑N (V rand mean(V rand )) 7. For each bus b / S, calculate the σ 2 = ℓ=1 ℓ − . b N 1 vk vk θk θk ∈ − 2 S r pk = Pr | |−| | < εv, − < εθ , k = 2,...,nbu, (51) 8. If σb < preset threshold for all b / , stop; else, continue. vk θk ∀ ∈ 2 ( ) 9. Move the meters in the set S to busesb with the largest variances σ . | | b 2 b 10. If required, add a meter on the bus with the largest σ ; update 1; where εv, ε are predefined thresholds. The goal is to obtain a probability b θ go to 2. Note that the algorithm could be modified to remove meters index pk > 0.95 for all but the substation bus, k = 1, where supposedly from the initial list of buses if the variance is smaller under all loadingb an accurate measurement device is already in place. From (51): conditions. b The previous method is independent of any state estimator, as vk µk = E | | , (52) opposed to the method in [177, 178], in which the problem definition is θk stated as follows. Find a measurement placement that makes the system   b T observable with established accuracy at a minimum cost. The accuracy vk vk requirement is a consequence of the use of pseudo-measurements with Rk = E b| | µk | | µk . (53) θk − θk − corresponding large uncertainties. For example, it is suggested in [179]      ! that the error in the pseudo-measurements is in between 20% and 50%. b b 2 Note that the error covariance matrix, Rk, associated with bus k is Formally, consider a system with n state variables, and define σmax, the b b maximum acceptable variance of any estimated state variable. Then, for two-dimensional. The authors rely on the geometric interpretation of the a given set of measurements: error covariance matrix—in particular, the fact that the two-dimensional n 2 error covariance matrix can be geometrically seen as an ellipse—as an σk indication to where measurements shall be allocated. The strategy is to min J = ∑ , place measurements at the locations where the ellipse area, proportional k=1 σmax (49)   to √detRk, is largest. The accuracy of the estimated state variables is s.t. σk σmax k = 1,...,n, improved by shrinking the ellipse areas. Ultimately, this is a way to ≤ ∀ reduce the variance of the estimates, as in previous work. But this idea where σk is the variance associated with the estimated state variable, xk. is nicely cast on a more formal statistical foundation in [179, 180]. The Finding the optimal solution to the combinatorial optimization problem algorithm in [179] is as follows: in (49) is challenging. A suboptimal solution to (49) is achieved in [177, 178] by using dynamic programming. The proposed method proceedsb as 1. Run the state estimator over a set of Monte Carlo simulations. follows. First, using the model of a distribution grid of interest: 2. For εv = 1%, εθ = 5%, if pk > 95%, k = 2,...,nbu, stop; else go to 3. 3. If the relative errors in voltage magnitude are within bounds, go to 5; else go to 4. 4. Place a voltage meter on bus k with the largest √detRk. 5. Compute the error covariance matrix corresponding to the real and ¶https://github.com/sedg/ukgds reactive power flow in each branch using the accordingly modified (53).

pp. 1–18 11 pmu sm a a a where ck ck denotes the cost of installing a PMU (smart meter) on bus k; βv and βθ are parameters that can be used to give different b b b weights to voltage
amplitude and phase deviations; and N is the number c c c of considered operating conditions; d d d pmu sm 1 if a PMU [smart meter] is installed on bus k, . . . uk [uk ]= . . . 0 otherwise, Initial w w w  (55) configuration x x x k vk vk k ev := max | |−| | ; and e := max θk θk . (56) y y y k v θ k − k | | z z z The optimization problemb (54) is solved by a genetic b algorit hm. Solu- tions that violate accuracy limits are penalized, thereby increasing their cost. Solutions with same costs are ranked by their accuracy. Finally, the Candidate location Selected location method finds a Pareto optimal front that finds a measurement placement solution for varying degrees of placement costs and estimation accuracy Fig. 5: Pictorial idea of the sequential method in [177, 178]. under several network configurations. This approach in [182] is extended in [184] to consider the uncertainties associated with DGs. Specifically, the DG outputs are used as pseudo-measurements with (non-Gaussian) unknown probability distribution functions modeled as Gaussian mixture 6. Place a power meter on branch ℓ with the largest √detRℓ;goto1. models. The work in [186] develops a PMU placement method that extends The algorithm proposed in [180] extends and improves on the previ- 1 the concept in [105] to distribution grids. The trace of G− (refer to ous algorithm. See also [189, 191]. A-optimality in Subsection 2.1.3) is the selected criterion to evaluate Following previous work, the variance of the estimated quantities the estimation accuracy. Following previous work [177–185, 187], the is the adopted criterion in [187] to evaluate the estimation accuracy; authors also resort to Monte Carlo simulations; this is because mea- however, the novelty in [187] is to link the gain matrix to a circuit rep- surement placement methods for distribution systems must account resentation. For example, nodal voltage measurements are represented for frequent topological reconfigurations. The formulated problem is by shunt admittances—branch current, power flow, and power injection solved by using a robust submodular optimization algorithm, referred measurements are also considered. The gain matrix is represented ana- to as a submodular saturation algorithm. It is demonstrated through lytically in the form of a network admittance matrix, which enables the numerical simulations that the submodular saturation algorithm outper- reformulation of the measurement placement problem as mixed-integer forms greedy and genetic algorithms in most cases. See also [195], linear programming with disjunctive inequalities. which focuses specifically on microgrids. Following [186], the works in Minimum number of measurement points for weak observability: [188, 190, 193] also approach the measurement placement problem from the standpoint of an optimal experimental design; only standard meters In all the previous work, the goal is to minimize the total number 1 are considered. The largest diagonal entry of G− is the selected (M- of measured quantities, i.e., the total number of measured voltages, cur- optimality) criterion to evaluate the estimation accuracy in [188, 190], rents, and powers. The authors of [185] argue that minimizing the total whereas a D-optimality criterion is selected in [193]. Finally, the work number of measured quantities across the grid might lead to a large in [194] explores properties such as convexity and modularity of differ- number of geographically spread measured points, thereby leading to ent metrics in the context of an optimal experimental design to propose the need of sensors at more locations and the resulting overall high and compare several tight lower and upper bounds on the performance of cost. Alternatively, they advocate that the optimization could focus on the optimal solution; the focus is exclusively on the placement of PMUs. ensuring the maximum utilization of available information from a few Measurement placement heuristics/formulations for static state esti- locations, i.e., it might be cost-effective to measure as many quantities mation in distribution grids rely heavily on pseudo-measurements, as possible from the same or a few locations; therefore, the goal here is whether or not the notion of weak numerical observability is considered. to minimize the total number of measured points while maximizing the Though convenient to address short-term goals, this strong dependence knowledge from each of those points—a goal that was also pursued by on pseudo-measurements is not effective as a long-term measurement [178] in identifying the use of more current measurements from the same placement strategy. This is further aggravated by distributed generation node. Apart from this philosophical difference, the measurement place- that adds another layer of uncertainty on pseudo-measurement models. ment method proposed in [185] follows along the same lines of previous The progressive elimination of high-uncertainty pseudo-measurements works, as summarized next: via a multistage formulation is a gap in the literature and an opportunity For all medium-voltage substations: for future developments. This is further discussed in Section 4. Place voltage measurements on each bus. • Place current measurements on each feeder leaving the substation. 3.2 Stability assessment and control in distribution grids • For all large industrial loads and DGs: 3.2.1 Voltage control in distribution grids: Place voltage measurements on each large industrial load and DG bus. Given that distribution system state estimation is still an emerging • Place current measurements on each branch connected to large indus- field, some studies have considered placing measurements of power to trial• load and DG buses. improve load flow calculations and the consequent voltage control strate- gies. A method for placing power flow measurements in low-voltage Further, at least one point of connection with household loads and one networks to improve power flow calculations in medium-voltage net- point of connection with commercial loads must be measured following works is proposed in [200, 201]. Along similar lines, but with a focus the scheme for large industrial loads and DGs. Finally, following pre- on volt-var control, [202] proposes a measurement placement strategy vious work, additional measurements should be placed to improve the for identifying the most important locations that can help to achieve the performance of the static state estimator; however, previous works use best performance of conservation voltage reduction. The algorithm starts the variance of voltage magnitudes and phase angles, estimated on those with an initial set of meters and estimates the voltage profiles. Using the nodes that are not measured, as a metric to evaluate the obtained results. standard deviation of estimated nodal voltages, the least significant mea- Conversely, [185] uses the variance of branch voltage phasors as a pri- surement is removed based on the voltage estimation accuracy, until one mary criterion and the variance of complex power flows as a secondary is left with the most significant locations for voltage measurements. criterion. The overall strategy in [185] seems to be supported by a Dutch Stability assessment and associated control schemes represent a new distribution grid operator and is well justified; the trade-offs between the paradigm in distribution grids. Accordingly, the number of works on proposed method and previous work are not elaborated. measurement placement with this focus is scarce. This scenario is rapidly Consideration of advanced metering infrastructure: changing with the integration of DERs to modern distribution grids. Except for microgrids operating in island mode, the focus on stability To this point, standard meters—that is, current, voltage, and power assessment and control for measurement placement in distribution grids meters—have been considered. But the next generation of distribution remains less relevant. grids will benefit from more advanced measurement systems [199], referred to as advanced metering infrastructure. Accordingly, a first 3.3 Topology change detection in distribution grids attempt to formulate the measurement placement problem for distri- bution grids to consider PMUs and smart meters, in addition to stan- Most previous work addresses the high degree of uncertainty in power dard meters, is reported in [182]. The problem is formulated as an distribution grids by relying on Monte Carlo simulations. An interesting optimization, and it follows closely the idea in [179]. Formally: departure from this trend is found in [196]. The authors recognize that nbu nbu N N the probabilistic approach based on Monte Carlo simulations depends min cpmu upmu csm usm β eℓ β eℓ ∑ k k + ∑ k k + v ∑ v + θ ∑ θ , on the statistics of measurement noise and pseudo-measurements, which k=1 · k=1 · ℓ=1 ℓ=1 are unknown and time-varying. Instead, the authors propose a deter- (54) ℓ ministic approach based on grid structural notions—namely, topology e εv, s.t. v ≤ detectability and outage identifiability—that depend only on the sys- eℓ ε , ℓ = 1,...,N, tem topology under normal operating conditions. Note, however, that  θ ≤ θ ∀

pp. 1–18 12 distributed generation is not considered. The goal in [196] is to find a measurement configuration to guarantee that topology changes can be Tu du T T = T d = , LT = (Lu La), (61) detected and identified. The problem is formulated as an optimization Ta → da | problem that ensures topology change detection at minimal cost. Given h i h i the definitions: where da and du contain variables associated with buses in which micro- PMUs are available and unavailable, respectively. It follows that: B: the set of all buses in the distribution grid • B Ludu = Lada. (62) k: the set of all buses downstream of bus k in a radial network − • E: the set of all branches in the distribution grid The key idea in [205] is to project (62) onto the subspace spanned • I 0: the set of all zero-injection buses in the distribution grid by the left singular vector, υu, corresponding to the smallest singular • bu ck : the cost of installing a meter on bus k value of Lu. It is expected that this procedure suppresses the effect of • br quantities associated with buses in which micro-PMUs are not avail- c(k,ℓ): the cost of installing a meter on branch (k,ℓ) • bu able, thereby making the anomaly detection framework a function of u = 1 if a meter is installed on bus k, and 0 otherwise available micro-PMU measurements only. In other words, in steady-state • k ubr = 1 if a meter is installed on branch (k,ℓ), and 0 otherwise operation, υHL d should be numerically small, and the normalized • k u u u dk: the degree of bus k, i.e., number of branches incident to bus k quantity: • H rk: the index of the bus immediately upstream of bus k, i.e., the parent υ L d bus.• u a a 2 , (63) Then: da bu bu br br k k min ck uk + c u , should vary smoothly in time. Consequently, anomalies are detected ubu ubr ∑ ∑ (k,ℓ) (k,ℓ) , k B (k,ℓ) E from abrupt variations in (63). Accordingly, the micro-PMU placement ∈ ∈ bu bu br is formulated as a min-max optimization problem, as follows: s.t. d1u1 + ∑ uℓ + ∑ u(1,ℓ) d1 1, H ℓ B (1,ℓ) E ≥ − opt da Xda ∈ 1 Π = min max , ∈ Π d 2 bu bu br a da dquq + ∑ uℓ + ∑ u(q,ℓ) (57) k k ℓ Bq (q,ℓ) E T T ∈ ∈ s.t. (Lu La)= L Tu Ta , B | | dq 2 q 1 having dq 3,   (64) ≥ − ∀ ∈ \{ } ≥ T = I2 (Π I3), ubu + ubr = 1 q I , ⊗ ⊗ q (rq,q) 0 H H ∀ ∈ X = La υuυu La, bu br E uℓ + u(k,ℓ) 1 (k,ℓ) . [Π] 0,1 , [Π] = 1, [Π] = 1. ≤ ∀ ∈ kℓ ∈ { } ∑ kℓ ∑ kℓ It is proven in [196] that the constraints in (57) guarantee topology k ℓ change identifiability; the solution to (57) is obtained by using a dynamic The formulation in (64) seeks to minimize the number of measure- programming algorithm. We remark that [194, 196] are, to the best of our ment devices while maximizing the range of buses where anomalies can knowledge, the only published works that present numerical results on be detected. The min-max optimization problem (64) is solved by using a distribution test systems of relevant size. See Table 5. greedy search. This approach has been recently extended to the problem of PMU placement for fault location [206]. 3.4 Fault detection in distribution lines 3.5 The work in [203] is the first to consider the problem of measurement Power quality monitoring in distribution grids placement—specifically PMUs—for fault location in distribution grids. A measurement placement method for power quality estimation based on Initially, one PMU is placed at the substation. The prefault and fault-on the notion of entropy is proposed in [207]. Monte-Carlo simulations are current phasors acquired by the PMU at the substation are used to calcu- used to draw samples of network states at metered and nonmetered loca- late the fault resistance and the fault-on voltage phasors. Then, additional tions. Then, Bayesian inference is used to obtain maximum-likelihood PMUs scattered at strategic locations provide fault-on voltage phasor estimates of the states at nonmetered locations. Based on simulated and measurements. These locations are chosen to minimize the error between estimated states at non-metered locations, the most poorly predicted non- the calculated and measured voltage phasors. Formally: metered locations are selected as the location for the next meter. See also [208]. nbu nbu min ∑ ∑ ukℓ k=1 ℓ=1 4 Discussion and avenues for future research ℓ=k 6 (58) Previous sections revisited and discussed the different methods for mea- surement placement in transmission and distribution grids. The methods nc (k) (ℓ) ukℓ = 1, if ∑q=1 vq vq = 0, were categorized by application. To this point, the paper provides a fairly s.t. − comprehensive description of the critical factors that go into formulating (ukℓ = 0, otherwise, a measurement placement problem. Although state estimation remains where: the central application, the paper also summarizes the use of mea- surement placement algorithms for other applications related to power nc: number of candidate locations for measurement placement • (k) system stability and online security assessment. Now, the objective of vq : measured voltage phasor on bus q for a fault at bus k this section is to provide a summary of what was discussed in previous • (ℓ) vq : calculated voltage phasor on bus q for a fault at bus ℓ. sections with a forward-looking view. • Allocation algorithms developed for state estimation problems gen- The optimization problem (58) is solved by using a Tabu search [203]. erally defined observability in a numerical or topological sense, and This method is extended in [204] to account for multiple fault scenarios past works typically tried to optimize the sensor allocation to either by using Monte Carlo simulations. In this case, instead of a Tabu search, maximize the observability against all uncertainties for a given budget a greedy randomized adaptive search metaheuristic is adopted. Notice and/or allowable estimation errors or minimize the cost of allocation that these methods [203, 204] are specifically designed to algorithms given a specific minimum requirement of observability and estimation that use voltage sag information to locate faults. errors. It was found that formulations that used the topological definition The work in [205] is mostly concerned with the detection of anoma- of observability—especially integer programming-based methods—lend lies in distribution grids using micro-PMUs. A comprehensive anomaly themselves well for incorporating uncertainties related to measurement detection framework is developed. The authors recognize that it is not errors, topological changes, and PMU or branch outages. The numerical feasible to deploy micro-PMUs on all system buses and that the per- observability-based problem formulation—though thorough in its con- formance of the developed framework depends on the number and sideration of improving state estimation accuracy—is not always easy location of available micro-PMUs; therefore, a micro-PMU placement to formulate. In practical situations, network topological information methodology specifically tailored to the anomaly detection framework may be lacking and/or the complexity of the optimization problem under is developed as follows. For a three-phase system of nbu buses, the uncertainties may be high. following algebraic equation holds true during steady-state operation: Other concepts that are considered in the measurement placement Ld = 0. (59) for state estimation improvement are critical measurements, measure- ment redundancy, and infrastructure costs. The concept of degree of where: T T T T T T T observability also provides a practical way to account for phased deploy- L = I3n Y ; d = i1 i2 i3 v1 v2 v3 ; (60) ment in the measurement placement formulation. Other critical factors bu |− considered in the placement problem for transmission systems are: I3n is the identity matrix; Y is theh three-phase admittancei matrix; and T bu T
i1 (v1 ) is a vector of the bus current injection (bus voltage) phasors, Cost of PMU: In addition to sensor cost, other costs included substa- associated with phase number 1. The symbol “|” is used to denote that tion• upgrade, instrument transformers such as CTs/PTs, communications the matrix L is composed by two sub-matrices; the top submatrix is costs, installation and foundation costs, network maintenance cost, data I3nbu , and the bottom submatrix is Y . Now, define a transformation archival platforms such as phasor data concentrators, and/or selected matrix T such that: − application costs of wide-area measurement and control systems

pp. 1–18 13 Network parameters: participation factor of system states, zero- Table 5 Distribution grid test systems • injection buses, preexisting PMU measurements System name No. buses Published works PMU aspects: measurement channel limits, data availability, PMU 11-Node 11 [208] outages,• degree of redundancy IEEE 13-Node 13 [172, 187, 207] Uncertainties modeled: load variations, generation or branch out- UKGDS #1 16 [182, 184] age/variations,• network parameters, DER outputs, transmission line Italy #2 17 [177] outages, topology changes, N-1/N-2 contingencies Italy #3 25 [200, 201] Other applications in addition to state estimation: dynamic state 30-Node 30 [163, 196] estimation,• stability assessments such as transient rotor angle stabil- 32-Node 32 [186] ity and voltage stability, line outage or topology detection, and fault IEEE 33-Node 33 [187, 192, 195, 198] identification. IEEE 34-Node 34 [168, 172–175, 202, 205, 206] IEEE 37-Node 37 [196] Some of these applications related to stability and security assess- Italy #4 51 [178] ments typically use the concept of identifying the most critical bus 55-Node 55 [188] (with respect to a certain stability or security index of interest). The IP IEEE 69-Node 69 [189, 191] method for formulating the measurement allocation problem can easily 70-Node 70 [186] accommodate such considerations of critical buses for sensor placement. UKGDS #2 77 [181, 182] In works related to distribution system measurement placement, the Netherlands 77 [185] typical strategy is to reduce the estimation errors for system states and India #3 85 [189, 191] Italy #5 84 [178] depends on the use of pseudo-measurements and Monte Carlo simu- Italy #6 95 [183] lations. This is partly because of the large node-to-measurement ratio UKGDS #3 95 [176, 179, 180, 190] in distribution systems and the general lack of observability—or even 119-Node 119 [186, 192] the lack of network metadata to model the state estimation problem IEEE 123-Node 123 [172, 194, 196, 206] using either the notion of numerical or topological observability. As a Brazil #4 134 [204] consequence, studies use a less strict definition of topological observabil- Brazil #3 136 [193] ity, and missing data are filled with pseudo-measurements or planning Brazil #5 141 [203] data. Given the uncertainties introduced because of the lack of data as 183-Node 183 [196] well as increasing levels of distributed resources, studies also resort to Europe #2 906 [196] probabilistic approaches to solve the estimation or stability assessment IEEE 8500-Node 8500 [194] problems; hence, Monte Carlo simulations provide a simple and straight- forward way to ensure that a measurement placement solution is suitable for a large percentage of time against operational uncertainties. Typi- cal factors considered in the distribution grid measurement placement problems are: micro-PMUs—must also be considered while analyzing which location will benefit from merging units data. Heterogeneous measurement devices (PMUs, micro-PMUs, smart meters),• multichannel measurements, pseudo-measurements Predefined number of meters, rule-based candidate selection Consideration of notions of stability other than voltage stability • Uncertainties: load time series, DG uncertainties, measurement device and• rotor angle stability: There is renewed interest in other notions of uncertainty,• network parameter, power flow uncertainties, topology stability, including frequency stability and resonance stability [138], due reconfiguration, pseudo-measurement uncertainties, critical data loss to the integration of converter-interfaced generation into modern power and associated costs. systems. These notions have never been considered in the measurement placement, and they represent an open field to be exploited. Reso- nance stability assessment, in particular, will require high-resolution 4.1 Future directions measurements provided by merging units. Consideration of leverage measurements for static state estima- • Elimination of pseudo-measurements for static state estimation tion in transmission grids: The notion of leverage measurement was • presented in Subsection 2.1.2. We conjecture that the IP method and in distribution grids: The reliance on pseudo-measurements has long its extensions can be modified to address the issues caused by lever- been the only alternative for static state estimation in distribution grids; age measurements. One approach is to use the hat matrix S given in however, the continuous increase in distributed generation is increas- (30). Recall that z = Sz. It follows that the diagonal element of the hat ing the level of uncertainty associated with pseudo-measurements. The elimination of pseudo-measurements has never been considered in the matrix, 0 < Sii < 1, represents the influence of the i-th measurement z on its estimate, z. This fact can be used to avoid placing leverage mea- measurement placement problem, and it could be an interesting direc- tion to pursue. The progressive elimination of pseudo-measurements via surements, that is,b to avoid placing measurements for which Sii is larger than a threshold. This approach is suggested in chapter 6.3 of [5]. How- a multistage formulation is of particular interest to the industry. ever, Sii is directlyb related to the Mahalanobis distance, hence vulnerable to the masking effect [209]. Another approach is to place a cluster of Open-source software: To date, open-source software for measure- measurements around locations known to create leverage measurements. •ment placement are not available. Power system operators are highly This approach is suggested in [96]. This remains a open research topic interested in such a computational tool, but in general they do not have and it requires further investigations. in-house expertise to develop it. Moreover, it is difficult to justify the investment in a tool that will not be continuously used. This type of Observability for dynamic state estimation in transmission grids: initiative must depart from governmental initiatives. Power• system dynamic state estimation is a timely topic. To achieve (strong) observability [130, 210] for dynamic state estimation represents Scalability and application to large realistic grids: The review pre- a major effort, a line of research that has been pursued [131, 133–135]. sented• in previous sections indicates that most measurement placement In all previous works, the synergy between static and dynamic state esti- work presented has been developed and tested on IEEE and other open- mation, briefly discussed in [211], has been completely neglected. Also, source test systems, with very few real system applications, e.g., the none of the previous works considered placing merging units in addition Italian network; see tables 5 and 6. The important information contained to phasor measurement units. An effective multistage strategy should in these tables is twofold: i) the performance of most existing meth- consider that modern digital substations [212] come with merging units ods has not been assessed on realistically sized systems; ii) research by standard. groups that have the capability to work with realistically sized systems are clearly identified. Merging units: Most work focuses on PMUs in transmission grids On transmission grid applications, several papers have applied their •and micro-PMUs and smart meters in distribution grids; however, emerg- proposed measurement placement algorithms on many test systems—the ing high-resolution sensing and measurement systems—such as merging IEEE 14-, 30-, 57-, and 118-bus systems are frequently used—thereby units that can acquire sample values at 5–10 kHz—will also play a vital providing a means to compare the algorithms and results; however, such role in future grids with increasing levels of dynamic and stochastic phe- a practice is less common on distribution grid applications partly because nomena. Especially when it comes to i) developing improved dynamic of the fewer related works and level of maturity of the research. In other models as well as substation protection and control systems, and ii) words, measurement placement for transmission grid applications has detecting high-impedance faults with higher levels of inverter-based gotten more attention over the years than distribution grid applications, resources that typically contribute very low fault currents. The availabil- which are increasing in relevance with increasing penetrations of DERs. ity of such high-resolution data and their fast communications and data In general, there is a dearth of studies on large-scale, realistic systems. analytics becomes highly attractive for developing autonomous, resilient One possible future direction could be to apply appropriate methods systems. Though many such dynamic phenomena are local, the question to synthetic, large-scale, realistic test systems developed as part of the of interest for the optimal placement of merging units will focus on iden- ARPA-E GRID DATA program [216]. Both the open-source transmis- tifying the most important locations or substations among the millions of sion [217] and distribution [218] data sets available are large enough and inverter-interconnected candidate nodes that will be key to have merging will be able to challenge the measurement placement algorithms. For units. The availability of other high-resolution sensing and measurement instance, the synthetic Bay Area distribution data set models more than systems—such as digital fault recorders, digital protective relays, and 4 million customers.

pp. 1–18 14 Table 6 Transmission grid test systems generation output. This idea can be further extended to form an optimal System name No. buses Published works placement of measurements considering both the grid and weather sens- 5-bus 5 [91, 93] ing devices. Further extension could include measurements in buildings, WSCC 9-bus 9 [62, 75, 87, 118, 131, 133, 134, 160] water delivery systems, and fuel and transportation infrastructures. For IEEE 14-bus 14 [16, 24, 29–32, 34–36, 38, 39, 41, 43– instance, the optimal placement of all-sky imagers [219] or pyranometers 50, 52–55, 58–63, 65–67, 69, 71, 73, will be essential to forecast the net load in futures with higher pene- 74, 76, 80, 82–84, 86, 93, 102–105, trations of distributed solar; hence, leveraging data from smart meters, 107, 109–111, 113, 119, 121, 125, 133, PMUs, and weather measurements will provide an economic solution for 150–154, 161, 162, 213, 213–215] grid-edge observability. The solution strategy could also include remote 18-bus 18 [161] sensing, such as satellite image processing. For both short-term oper- Longitudinal 21-bus 21 [146] ational states and long-term planning states, learning algorithms could Southern Italy 22 [30] be trained to estimate the grid-edge states and identify the most criti- IEEE 24-bus 24 [41, 47, 48, 54, 150] West Bengal 24 [122] cal locations that will need sensing of appropriate electrical or weather New Zealand 27 [162] parameters to ensure the system observability, estimate the states, and Georgia-Florida 29 [147] inform stability indices of interest. One aspect related to this direction of Hydro-Québec #1 29 [148] work is the need for centralized processing systems that can assimilate IEEE 30-bus 30 [29, 35, 36, 39–42, 44–48, 50–55, 59, and curate all these disparate data streams and distribute them to various 61, 62, 67, 71, 82, 84–86, 102–104, operations or use cases in a business system. Some utilities have already 107–111, 113, 114, 119–121, 124, begun the development of a such a centralized data ingestion platform. 125, 150, 152, 153, 161, 213, 213] The use of the Apache Spark or Kafka platforms is also gaining trac- Central Southern Italy 38 [30] tion among researchers and the power industry to develop a scalable, New England 39-bus 39 [16, 28, 30, 31, 35, 37, 41, 45, 47–50, heterogeneous data ingestion and distributed analytics platform. 54, 55, 62, 67, 74, 78, 83, 102, 104, 108, 133, 140, 143, 145, 157] Iran #1 50 [153] Artificial intelligence, data anomaly detection and fusion with IEEE 57-bus 57 [24, 29, 31, 35, 36, 39–47, 52, 53, 55, • 59, 61–63, 65–67, 69, 71, 75, 77, 86, model-generated data: In addition to fusing raw measured data, arti- 87, 90, 91, 102, 107, 108, 111, 118, ficial intelligence techniques can add further richness to the data by 124, 125, 152, 213, 213, 214] using offline model-generated data, especially to capture the influence Brazil #1 61 [38] of low-probability, high-impact events and detect unforeseen patterns, Hydro-Québec #2 67 [148] especially for applications that are geared toward detecting anomalous New England 68-bus 68 [135, 149, 157] data, cybersecurity intrusions, and ensuring grid resilience—something North Central U.S. and Canada 75 [158] for which not many past works have designed optimal measurement Simplified Italy 76 [30] placement methods, as shown from the reviews in Section 2 and Section RTS96 96 [76, 87, 107, 121] 3. Future work must look to fuse disparate measurement system data IEEE 118-bus 118 [16, 24, 28, 30, 35–37, 39–47, 51, 55, sets along with validated model-generated data through artificial intelli- 60–69, 71, 78, 82, 86, 90, 91, 104, 105, 108–111, 113, 119–121, 123– gence. Typically information content (or entropy) is high in the region 125, 133, 152, 153, 215] of system parameter measurements where the variability of grid relia- Italy #1 129 [30] bility or resilience is high, e.g., the boundary region between acceptable NPCC 48-machine 140 [134] and unacceptable grid performance [7]. Given that such regions are less IEEE 50-machine 145 [135] represented in typical grid measured conditions, analytics that will fur- WSCC 173-bus 173 [16, 30, 32] ther enrich the data for such conditions through extensive offline model Mexico #1 190 [149] analysis will be needed. Such efficient sampling of training data can Taiwan #1 199 [16, 28] ensure optimal measurement placement for applications targeted toward Iran #2 242 [35] ensuring grid resilience. India #1 246 [34, 74, 83, 140] In terms of methodologies, typical methods for optimal measurement Taiwan #2 265 [16] placement are optimization- or heuristic-based. The former is highly USA #1 270 [52, 53] useful considering that planning and state estimation problems can be Central America 283 [68] represented in terms of optimization; however, considering the future 298-bus 298 [54, 71] research directions in this area that will need to work with large-scale, IEEE 300-bus 300 [41, 42, 47, 48, 51, 65, 68, 86, 108, 111, 113, 120, 124] realistic systems, detailed models of AC power flow equations, as well AEP utility company 360 [139] as voltage control devices, and that will need to consider both short-term USA #2 444 [52, 53] and long-term uncertainties and exploit the spatial and temporal syner- Denmark 470 [48] gies across multiple types of measurements (e.g., sensing for weather, Iran #3 529 [64] grid, building, asset health), scalable data-driven approaches that can India #2 996 [68] fuse the disparate data from myriad measurements are definitely needed. Europe #1 1354 [108] Machine learning and statistical methods, such as clustering and decision Brazil #2 1495 [125] trees [7, 220], as well as signal processing techniques [127] have already Entergy 2285 [84, 85] been used in the past to identify the most representative or influential Poland #1 2383 [41, 44, 55, 62, 63, 104, 108, 120] network nodes to be monitored and their related measurements. Such Poland #2 2746 [39, 44] methods are attractive especially for distribution grid applications [221], Poland #3 3120 [111] given the lack of network models and metering infrastructure, including Poland #4 3375 [65] on the secondary side of service transformers. USA #3 4520 [86] Mexico #2 5449 [145] USA #4 8000 [93] Business value proposition: For real utility or industry adoption of solutions,• the value proposition of measurement placement solutions will need to consider relevance to multiple stakeholders, applications, and planning time horizons. The papers reviewed typically focus on one or Planning uncertainties: Other areas of future research can be in two applications, and many revolve around ensuring complete or maxi- •terms of modeling several sources of power system uncertainties. Typ- mizing observability; however, a comprehensive placement method for ical considered uncertainties include PMU outage, branch outage, bad a business must consider myriad high-impact applications or use cases measurements, and load and DER outputs. Most of these are short-term (some of which were delineated in this paper) and also consider values operational uncertainties; however, given that the measurement place- from both short-term (grid operational performance) and long-term per- ment problem is a planning problem, long-term uncertainties related to spectives. Given that business models in utilities are evolving around the grid futures could also be considered, such as customer adoption levels use and management of data to ensure reliable service and innovative of DERs, variations in DER sizing, displacement of conventional syn- products for increased customer satisfaction, the measurement system chronous generators, degree of proliferation of smart inverters or control allocation problem must consider the value streams and applications devices, and load growth (including electric vehicle penetrations). These important for a typical business model. One way to ensure this will be to factors can dictate the required level of observability and controllability work on real system data and to work closely with industry stakeholders needed for various applications, the stability drivers in the grid, and will to exploit the data streams, both live and historical archives. consequently influence the locations, type, and number of measurements required.

Synergies from disparate measurement systems and scalable pro- 5 Conclusions cessing• systems: Another future direction to pursue is modeling and This paper reviewed various methodologies for optimal measurement quantifying the synergies across multiple measurement systems. Most placement in transmission and distribution systems. In general, all the work reviewed in this paper considered synergies across grid electrical transmission/distribution systems have preexisting measurements, but output measurements, e.g., PMUs, meters. Very few work has also con- most methods ignored this fact and placed measurements afresh, and sidered weather forecasts as pseudo-measurements of variable renewable very few methodologies considered preexisting measurements. Future

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